On symmetric association schemes and associated quotient-polynomial graphs
Rights accessOpen Access
All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder
ProjectESTUDIO MATEMATICO DE LOS FALLOS EN CASCADA EN SISTEMAS COMPLEJOS MEDIANTE INVARIANTES Y CENTRALIDADES EN GRAFOS. APLICACIONES A REDES REALES. (AEI-PGC2018-095471-B-I00)
Let denote an undirected, connected, regular graph with vertex set , adjacency matrix , and distinct eigenvalues. Let denote the subalgebra of generated by . We refer to as the adjacency algebra of . In this paper we investigate algebraic and combinatorial structure of for which the adjacency algebra is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) has a standard basis ; (ii) for every vertex there exists identical distance-faithful intersection diagram of with cells; (iii) the graph is quotient-polynomial; and (iv) if we pick then has distinct eigenvalues if and only if . We describe the combinatorial structure of quotient-polynomial graphs with diameter and distinct eigenvalues. As a consequence of the techniques used in the paper, some simple algorithms allow us to decide whether is distance-regular or not and, more generally, which distance- matrices are polynomial in , giving also these polynomials.
CitationFiol, M.; Penjic, S. On symmetric association schemes and associated quotient-polynomial graphs. "Algebraic Combinatorics", 1 Gener 2021, vol. 4, núm. 6, p. 947-969.